Finite-sheeted covering spaces and a Near Local Homeomorphism Property for pseudosolenoids

Fox theory of overlays is used to characterize finite-sheeted covering spaces of pseudosolenoids (i.e. hereditarily indecomposable circle-like nonchainable continua). Let S be a P  -adic pseudosolenoid, for a sequence of prime numbers P=(p1,p2,…,pn,…)P=(p1,p2,…,pn,…). S admits a d-fold cover onto itself if and only if d   is relatively prime to all but finitely many pipi. Moreover, if C is any connected finite-sheeted covering space of S then C is homeomorphic to S  . These results parallel known results for solenoids and extend results of Bellamy and Heath for the pseudocircle. In addition, it is shown that most self-maps of pseudosolenoids are finite-sheeted covering maps. Namely, if E(S)E(S) is the space of all surjective self-maps of S   with the sup metric, then the subset C(S)C(S) of local self-homeomorphisms of S   is a dense GδGδ in E(S)E(S). This result, based on a theorem of Kawamura, extends the near-homeomorphism property of the pseudoarc proved independently by Lewis and Smith, and relates to a question raised by Lewis in 1984, as to which other nondegenerate continua have the near-homeomorphism property.